If a irreducible, stochastic, aperiodic matrix is not diagonalizable, can it converge on power method??

Question

In some linear algebra books/blogs, I have found that a power method can be done only on diagonalizable matrix with a dominant eigenvalue. In other books/blogs, I have found that a irreducible, stochastic, aperiodic matrix can converge on the power method. I am working with irreducible, stochastic, aperiodic and it may be diagonalizable sometime or not sometimes. I want to know if it would always converge on the power iteration method. I also want to know why a non-diagonal matrix with a dominant eigenvalue can not converge?

Answer 1

The power method can be used to analyze matrices that are not diagonalizable. The way to do it is using a similar transformation, called Jordan decomposition (see here), where unlike the eigenvalue decomposition, Jordan decomposition always exists.